For numerical integration, a quadrature formula

$$\int_{a}^{b} f(x) d x \approx \sum_{i=0}^{n} a_{i} f\left(x_{i}\right)$$

is applied which is exact on $\mathcal{P}_{n}$, i.e., for all polynomials of degree $n$.

Prove that such a formula is exact for all $f \in \mathcal{P}_{2 n+1}$ if and only if $x_{i}, i=0, \ldots, n$, are the zeros of an orthogonal polynomial $p_{n+1} \in \mathcal{P}_{n+1}$ which satisfies $\int_{a}^{b} p_{n+1}(x) r(x) d x=0$ for all $r \in \mathcal{P}_{n}$. [You may assume that $p_{n+1}$ has $(n+1)$ distinct zeros.]